Notice in the game \(\mathcal {G}^n\)  the attacker and the detector adopt sequential rationality to design their optimal strategies [1]}. The formulations (REF ) and (REF ) characeterize the detector's optimal decision rules. For the attacker's optimal strategies, i.e. optimal distorted density matrices \(\rho ^{(n)*}_0,\rho ^{(n)*}_1\in S^{\otimes n}\) , we state the following theorem.
[Attacker's optimal strategies]

Let \(\mathcal {G}\)  be the Stackelberg game between the attacker and the detector (the passive quantum detector) as mentioned in definition REF . Then we can derive the attacker's optimal strategies, characterized by distorted density operators \(\rho ^{\prime (n) *}_1, \rho ^{\prime (n) *}_0 \)  as follows:
\(\rho ^{(n)*}_0&=\rho ^{\otimes n}_0, \\\rho ^{(n)*}_1&= \frac{\exp (\ln \rho ^{\otimes n}_1 - \frac{1}{\beta _n}\Pi ^{n*}_1) }{\text{Tr}(\exp (\ln \rho ^{\otimes n}_1 - \frac{1}{\beta _n}\Pi ^{n*}_1))}.\) 
